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Perturbation Analysis of Parametric Resonance

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Parametric resonance arises in mechanics in systems with external sources of energy and for certain parameter values. Typical examples are thependulum with oscillating support and a more specific linearization of this pendulum, the Mathieu equation in the form

$$ \ddot{x} + (a + b \cos (t)) x = 0\:. $$

The time-dependent term represents the excitation. Tradition has it that parametric resonance is usually not considered in the context of systems with external excitation of the form \( { \dot{x}= f(x) + \phi(t) } \), but for systems where time-dependence arises in the coefficients of the equation. Mechanically this means usually periodically varying stiffness, mass or load, in fluid or plasma mechanics one can think of frequency modulation or density fluctuation, in mathematical biology of periodic environmental changes. The term ‘parametric’ refers to the dependence on parameters and certain resonances arising for special values of the parameters. In the case...

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Abbreviations

Coexistence:

The special case when all the independent solutions of a linear, T-periodic ODE are T-periodic.

Hill's equation:

A second order ODE of the form \( { \ddot{x} + p(t) x = 0 } \), with \( { p(t)\,T } \)-periodic.

Instability pockets:

Finite domains, usually intersections of instability tongues, where the trivial solution of linear, T-periodic ODEs is unstable.

Instability tongues:

Domains in parameter space where the trivial solution of linear, T-periodic ODEs is unstable.

Mathieu equation:

An ODE of the form \( \ddot{x} + (a + b \cos (t)) x = 0 \).

Parametric resonance:

Resonance excitation arising for special values of coefficients, frequencies and other parameters in T-periodic ODEs.

Quasi-periodic:

A function of the form \( { \sum_{i=1}^n f_i(t) } \) with \( { f_i(t)\,T_i } \)-periodic, n finite, and the periods \( { T_i } \) independent over R.

Sum resonance:

A parametric resonance arising in the case of at least three frequencies in a T-periodic ODE.

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Acknowledgment

A number of improvements and clarifications were suggested by the editor, Giuseppe Gaeta. Additional references were obtained from Henk Broerand Fadi Dohnal.

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Authors and Affiliations

  1. Mathematisch Instituut, University of Utrecht, Utrecht, The Netherlands

    Ferdinand Verhulst

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  1. Ferdinand Verhulst
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Verhulst, F. (2009). Perturbation Analysis of Parametric Resonance. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_393

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